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Properties

Label 2-2240-280.139-c1-0-45
Degree $2$
Conductor $2240$
Sign $0.485 - 0.874i$
Analytic cond. $17.8864$
Root an. cond. $4.22924$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.97·3-s + (−2.14 + 0.615i)5-s + 2.64·7-s + 5.82·9-s + 7.17i·13-s + (−6.38 + 1.82i)15-s + 6.81i·19-s + 7.86·21-s − 7.48·23-s + (4.24 − 2.64i)25-s + 8.40·27-s + (−5.68 + 1.62i)35-s + 21.3i·39-s + (−12.5 + 3.58i)45-s + 7.00·49-s + ⋯
L(s)  = 1  + 1.71·3-s + (−0.961 + 0.275i)5-s + 0.999·7-s + 1.94·9-s + 1.98i·13-s + (−1.64 + 0.472i)15-s + 1.56i·19-s + 1.71·21-s − 1.56·23-s + (0.848 − 0.529i)25-s + 1.61·27-s + (−0.961 + 0.275i)35-s + 3.41i·39-s + (−1.86 + 0.534i)45-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 - 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.485 - 0.874i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (1119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ 0.485 - 0.874i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.014170344\)
\(L(\frac12)\) \(\approx\) \(3.014170344\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (2.14 - 0.615i)T \)
7 \( 1 - 2.64T \)
good3 \( 1 - 2.97T + 3T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 7.17iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 6.81iT - 19T^{2} \)
23 \( 1 + 7.48T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 13.9iT - 59T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 5.65iT - 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 16.9iT - 79T^{2} \)
83 \( 1 - 13.8T + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.938934746012952925728984064680, −8.279294877207670562567010531098, −7.86868867150941157249214916788, −7.20216163360282101248848546869, −6.28719307232312207149831374420, −4.76806103913741951666259687215, −3.97464534213806632464219840294, −3.64154745470473515336335885259, −2.24326225286291344652739758984, −1.68312426272319101630890056512, 0.855290623212016635208127456367, 2.25441993749478404127380016352, 3.04347383084306922507259965185, 3.88614032590276612318826035425, 4.64019439855049141768763903102, 5.54714752418817373323062077119, 7.06173087507755644877804470673, 7.69445885757919336903979270863, 8.214315913393274555849365823442, 8.562088921767566222664194249890

Graph of the $Z$-function along the critical line